Yau's conjecture on the first eigenvalue
# Yau's conjecture on the first eigenvalue

In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks:

   Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ is $n$?

If true, it will imply that the area of embedded minimal hypersurfaces in $S^{3}$ will have an upper bound depending only on the genus.

Some possible reformulations are as follows:

The Yau's conjecture is verified for several special cases, but still open in general.

Shiing-Shen Chern conjectured that a closed, minimally immersed hypersurface in $S^{n+1}$(1), whose second fundamental form has constant length, is isoparametric. If true, it would have established the Yau's conjecture for the minimal hypersurface whose second fundamental form has constant length.

A possible generalization of the Yau's conjecture:

   Let $M^{d}$ be a closed minimal submanifold in the unit sphere $S^{N+1}$(1) with dimension $d$ of $M^{d}$ satisfying $d\geq {\frac {2}{3}}n+1$. Is it true that the first eigenvalue of $M^{d}$ is $d$?
